Here is how I rename a tag old to new:

git tag new old
git tag -d old
git push origin new :old

The colon in the push command removes the tag from the remote repository. If
you don’t do this, Git will create the old tag on your machine when you pull.
Finally, make sure that the other users remove the deleted tag. Please tell
them (co-workers) to run the following command:

git pull --prune --tags

Note that if you are changing an annotated tag, you need ensure that the
new tag name is referencing the underlying commit and not the old annotated tag
object that you’re about to delete. Therefore, use git tag -a new old^{}
instead of git tag new old (this is because annotated tags are objects while
lightweight tags are not, more info in this answer).